Lemma 51.17.2. Let A be a ring. If f_1, \ldots , f_{r - 1}, f_ rg_ r are independent, then f_1, \ldots , f_ r are independent.
See [Lech-inequalities] and [Lemma 1 page 299, MatCA].
Proof. Say \sum a_ if_ i = 0. Then \sum a_ ig_ rf_ i = 0. Hence a_ r \in (f_1, \ldots , f_{r - 1}, f_ rg_ r). Write a_ r = \sum _{i < r} b_ i f_ i + b f_ rg_ r. Then 0 = \sum _{i < r} (a_ i + b_ if_ r)f_ i + bf_ r^2g_ r. Thus a_ i + b_ i f_ r \in (f_1, \ldots , f_{r - 1}, f_ rg_ r) which implies a_ i \in (f_1, \ldots , f_ r) as desired. \square
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